60. Two Methods
c. Verify that your answers to parts (a)...

Question

60. Two Methods

c. Verify that your answers to parts (a) and (b) are consistent.

Accepted Answer

Welcome back, everyone. In this problem, we want to evaluate the integral of T multiplied by LN of 3 T squared with respect to T without using substitution. No Notice here that we have our product and if we're not going to use substitution, then we can integrate by parts. Recall that in integrating by parts if we're finding the integral of UDV we can rewrite it as UV minus the integral of VDU. So we can split our integral up in this way to solve, but the tricky part is determining which function to call you and which function to call DV. To help us do that, we can use the acronym Lipit, which stands for logarithmic functions, inverse trigonometric, polynomial, exponential, and trigonometric. So we can choose which function to call you based on this order. Now notice in our expression, we have a logarithmic function, LN of 3 T squared, and by order that comes first. L comes first. So that is the function that we would let be equal to you. So if we let you be equal to LN of 3 T squad. Then that means we would let dV be equal to T. I know if. U equals LN of 3 T squared, then we can go ahead and find the derivative of U with respect to T. Now notice here that we could rewrite you let me make some space actually. We could rewrite you, OK, as Ellen 3, OK. Plus LN of T squared using the properties of logarithms and now we could take it even further to rewrite it as LN 3 + 2 LNT OK, based on what we know about pores in logarithms we can bring down our report. Now that we have you equal to this expression, then if we differentiate you with respect to T, we'll find that the derivative of LN3 will be zero since it's a constant, and the derivative of 2 LNT. Will be equal to 2 divided by T. And if DU D T equals 2 divided by T, then that means DU is going to be equal to 2 divided by T multiplied by D T. On the other hand, if DV equals T, then if we integrate both sides, V will be equal to a half of T squared. Know that we have expressions for D U and V, then we can apply our integration by parts because no, that means then the integral of TLN of 3 T squared with respect to T is going to be equal to UV that's LN 3 T squared multiplied by a half of T squared, OK. Minus the integral of VDU. So that's minus the integral of T square divided by 2 multiplied by 2 divided by TDT. No, we can go ahead and simplify here, OK, because for a first term we can rewrite it as a half of T squared multiplied by the LN of 3 T squared. And for our second term, notice here that 2 cancels 2. T goes into T2 T times. So this leaves us with the integral of T with respect to T. And the integral of T with respect to T is equal to a half of T squared. So now, we could rewrite this as a half of T squared, LN 3 T squared. Minus 1/2 of T2, OK, plus C. And now if we factor out a half of T2. Then that leaves us with the LN of 3 T square minus 1, OK, plus C. Thus, this is the integral of TLN 3 T squared with respect to T. Thanks a lot for watching everyone. I hope this video helped.

60. Two Methods
c. Verify that your answers to parts (a) and (b) are consistent.

Key Concepts

Verification of Solutions

Multiple Solution Methods

Consistency in Mathematical Results

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